Inversion-like and Major-like Statistics of an Ordered Partition of a Multiset

Abstract

Given a partition λ = (λ1, λ2, . . . , λl) of a positive integer n, let Tab(λ, k) be the set of all tabloids of shape λ whose weights range over the set of all k-compositions of n and OPλrev the set of all ordered partitions into k blocks of the multiset {1l2l−1 · · · l1}. In [2], Butler introduced an inversion-like statistic on Tab(λ, k) to show that the rankselected Möbius invariant arising from the subgroup lattice of a finite abelian p-group of type λ has nonnegative coefficients as a polynomial in p. In this paper, we introduce an inversion-like statistic on the set of ordered partitions of a multiset and construct an inversion-preserving bijection between Tab(λ, k) and OP λ̂ . When k = 2, we also introduce a major-like statistic on Tab(λ, 2) and study its connection to the inversion statistic due to Butler. 1. Ordered Partitions of a Multiset Let n be a positive integer. An ordered partition of [n] := {1, 2, . . . , n} is a disjoint union of nonempty subsets of [n], and its nonempty subsets are called blocks. Conventionally we denote by π = B1/B2/ · · · /Bk an ordered partition of [n] into k blocks, where the elements in each block are arranged in the increasing order. The set of all ordered partitions of [n] into k blocks will be denoted by OPkn. In the exactly same manner, one can define an ordered partition of a finite multiset. The set of all ordered partitions of a multiset S will be denoted by OPkS . In particular, in case where S is a multiset given by {1, · · · , 1 } {{ } c1−times , 2, · · · , 2 } {{ } c2−times , · · · · · · , l, · · · , l } {{ } cl−times }, (simply denoted by {1122 · · · ll}), we write OPk(c1,··· ,cl) for OP k S . For each π = B1/B2/ · · · /Bk ∈ OP k S , the type of π is defined by a sequence (b1(π), b2(π), · · · , bk(π)), where bi(π) is the cardinality of Received July 29, 2013; revised March 17, 2014; accepted April 11, 2014. 2010 Mathematics Subject Classification: 05A17, 05A18, 11P81.